Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 119

A similar rearranging of approximate solution for the electric field is found in
Appendix A. In this case Eq. ( 4.33 ) should be replaced by the following equation:

X

iMF .!/
4" 0 R e d

.! C icZ=d/.2n C 1/
! 2 E r D

C ic!Z.!/=d ! n

P n .cos /: (4.38)

nD0

By symmetry of the problem the electromagnetic field of the vertical CG
lightning discharge is azimuthally symmetric in the Earth-Ionosphere cavity. Before
discussing these solutions, we note that Eq. ( 4.36 ) may be simplified in an extreme
case kR e 1 or ! ! 1 . Letting formally ! ! 0, we come to

X

1

0 MF .!/
4R e d

.2n C 1/
n.n C 1/ P n .cos /:

B ' D @ (4.39)

n D 1

The sum of infinite series in Eq. ( 4.39 ) is given by Eq. ( 4.61 ). Taking this sum and
performing differentiation over yields

cot
2

:

0 MF .!/
4R e d

B ' D (4.40)

This equation coincides with the known solution for quasi-steady magnetic field
of the radial point current element located inside the thin .d R e / spherical
condenser (e.g., see Belyaev et al. 1989 ). In the same approximation one can
simplify Eq. ( 4.38 )forE r .
The measured resonance frequencies, i.e., the frequencies at which the spectra
tend to maximize, undergo a weak diurnal variation with magnitude about ˙ 0:5 Hz
about their average values (Balser and Wagner 1962 ). Furthermore, the resonance
frequencies of the magnetic variations are observed to depend on the orientation
of the antenna; that is, these values depend on whether the measurement is made
along the north-south or east-west direction (Sentman 1987 ). The shift of the
eigenfrequencies of the Earth-Ionosphere cavity may occur due to changes in
dissipative properties of the ionosphere. The solar Roentgen radiation and energetic
particle precipitation from the magnetosphere may greatly affect the ionosphere
conductivity due to ionization of the neutral particles (Mitra 1974 ; Rodger 1999 ).
It should be noted that the shift in the eigenfrequencies is also expected for some
kinds of field asymmetry in the actual Earth-Ionosphere cavity. The electromagnetic
field of the vertical CG lightning discharge given by Eqs. ( 4.36 ) and ( 4.38 )is
azimuthally symmetrical in the Earth-Ionosphere cavity. The n-th normal mode of
this field is described by .2n C 1/-fold degenerate eigenfunction. This implies the
degeneracy of the eigenfunction with respect to azimuthal variable '. The observed
effect can be due to the removal of this degeneracy of the normal modes of the
resonator possibly due to some kind of accidental factors (e.g., see a topic by
Nickolaenko and Hayakawa ( 2002 )).